Does anyone know a reference treating the deformation theory of nonconnective commutative rings?

Hello! Wondering if anyone can point me in the right direction about this: I want to learn some modern accounts on homotopy theory and derived version of classical theories, probably those in and related to the new released Handbook. Naturally, I know there are references like Lurie, but time and background knowledge are playing against me.

Ideally I'd like to arrange some kind of one semester course long material for myself. So perhaps 70 pages like the link comes a bit short of that.

@CharlesRezk, hello! I work with Vladimir Dotsenko (he's in Strasbourg now, but originally in Trinity College Dublin). I was reading your thesis at some point, where you prove that Hochschild cohomology of an algebra over an operad (if I recall correctly) relates to operadic cohomology by a 4 term exact sequence and a shift.

And a LES.

I wasn't able to follow your arguments mostly from my own ignorance of some language you use.

But I think that should be the idea?

I think that is the idea, yes.

Let $ \mathrm{Cat}_\infty$ denote the (large) quasicategory of small quasicategories, and let $\mathrm{Cat}_{(n,1)}$ denote the full subcategory spanned by $(n+1)$-coskeletal quasicategories. Is it known whether or not $\mathrm{Cat}_{(n,1)}$ is $(n+2)$-coskeletal? It seems like something that might be true, but I'm having trouble finding a reference.

@BrianShin I've claimed before that a quasicategory is $(n+1)$-coskeletal if and only if it models an $(n,1)$-category, though I don't know a reference for this fact. Assuming I'm not mistaken, one could then appeal to the fact that $(n,1)$-categories are the same as categories enriched in $n$-types. Then the homspaces in a functor category between two such is a certain limit of $n$-types

and so also an $n$-type

Anyway, I'd agree that the answer has to be yes, but I also agree it might be tricky to piece together a proof from the literature.

@TimCampion Right, this is exactly the idea I'd like to have pinned down.

@TimCampion What do you mean by "models an $(n,1)$-category" here?

@CharlesRezk Thanks a bunch.

Would anyone have any suggestions regarding my Q above?

@AlexanderCampbell I just mean that the hom-spaces are $n$-truncated.

@AlexanderCampbell Ah, I think I see what you're getting at

being $(n+1)$-coskeletal is not a condition that's invariant under equivalence

but modeling an $(n,1)$-category is

so indeed my statement was too sweeping

@TimCampion Right

I suppose the correct thing to say is that if $X$ is a quasicategory, then the map from $X$ to its $(n+1)$-coskeletalization is an equivalence iff $X$ models an $(n,1)$-category.

But maybe that's not quite supported by the argument I sketched in that MO question

looking back, I think this more modest statement is indeed supported

@TimCampion Edoardo and I proved this in this paper: arxiv.org/abs/1810.11188 (this link doesn't work at the moment because the arXiv is currently down). The argument is similar to your MO answer (which I hadn't seen until now).

Awesome!

I should say that the other thing you might have meant by "models an $(n,1)$-category" is that the quasi-category is an "n-category" in the sense of HTT2.3.4. In that case, $(n+1)$-coskeletality is necessary but not sufficient.

Interesting! I think that's a section of HTT I haven't looked at before!

Duskin's paper tac.mta.ca/tac/volumes/9/n10/9-10abs.html goes into great detail in the low dimensional cases.

Oh cool! I guess this is the paper introducing the Duskin nerve, too. I've been wondering about that a bit lately -- in particular I've been wondering whether it's been shown to "model the correct thing"

Martina Rovelli and Viktoriya Ozornova recently had a paper about this arxiv.org/abs/1902.05524 (N.B. arXiv is still down...)

And Edoardo and friends had a paper up the other day which studies (among other things) the analogous nerve landing in scaled simplicial sets.

Wow, that's great! I really need to be better about reading things.

(A link to the latter: arxiv.org/abs/1911.01905)

Unrelated equivariant question: Let $G$ be a compact Lie group and $H \subseteq G$ a subgroup. What do you call a genuine pointed $G$-space $X$ whose $K$-fixed points are contractible for $K \neq H$? The category of such is equivalent to the category of pointed Borel $W$-spaces, where $W$ is the Weyl group of $H$ in $G$, but I imagine there should be a name for the inclusion functor or something.

Er -- "$K \neq H$" should be "$K$ not conjugate to $H$" of course

@TimCampion Usually you say that the pointed space is "concentrated at $H$"

Sweet, thanks!